U n i v e r s i t y of H e i d e l b e r g
Discussion
Paper
Series
No.
420
Department
of
Economics
Herding with and without Payoff
Externalities -An Internet Experiment
Mathias Drehmann, Jörg OechsslerHerding with and without Payo¤ Externalities
-An Internet Experiment
Mathias Drehmanny Bank of England Jörg Oechsslerz University of Heidelberg Andreas Roider University of Bonn and Stanford University November 2004 AbstractMost real world situations that are susceptible to herding are also characterized by direct payo¤ externalities. Yet, the bulk of the theoretical and experimental literature on herding has focused on pure informational externalities. In this paper we experi-mentally investigate the e¤ects of several di¤erent forms of payo¤ externalities (e.g., network e¤ects, …rst-mover advantage, etc.) in a standard information-based herding model. Our results are based on an internet experiment with more than 6000 subjects, including a subsample of 267 consultants from an international consulting …rm. We also replicate and review earlier cascade experiments. Finally, we study reputation e¤ects (i.e., the in‡uence of success models) in the context of herding.
JEL–classi…cation: C92, D8.
Key words: information cascades, herding, network e¤ects, experiment, internet.
We are grateful to Jeremy Bulow, Muriel Niederle, Garth Saloner, Katja Seim, Catherine Tucker, and Je¤rey Zwiebel for helpful conversations, and seminar participants at Stanford for comments. We thank Frauke Lammers, Susanne Niethen, and Nina Wessels for very useful discussions throughout the project, and McKinsey & Company for …nancial support. Financial support was also provided by SFB/TR 15, GESY. The third author gratefully acknowledges the hospitality of Stanford Graduate School of Business and …nancial support by the German Research Foundation. Finally, we are very grateful to Ben Greiner who did an excellent job programming the experiment.
yThe views expressed in this paper are those of the authors and do not necessarily re‡ect those of the
Bank of England.
zCorresponding author: Department of Economics, University of Heidelberg, Grabengasse 14, 69117
1
Introduction
Motivation Whether one considers investment decisions, bank runs, fashion markets,
or even the choice of restaurants, herding behavior seems to be ubiquitous in human de-cision processes. Despite the bad reputation that herding behavior sometimes enjoys, it can be justi…ed as a rational response to uncertainty and informational asymmetries in the environment. Several sources of rational herding have been described in the theoretical literature. Information cascade models, pioneered by Bikhchandani, Hirshleifer and Welch (1992), Welch (1992), and Banerjee (1992), show that herding may occur even when in-dividuals’ payo¤s do not in any way depend on the behavior of others.1 In these models
externalities are created only through the information that can be deduced from observed actions.2 Other explanations of herd behavior are based on payo¤ externalities, which seem to be widespread in practice. For example, herding of analysts or fund managers in models of reputational herding (e.g., Scharfstein and Stein, 1990), or herd behavior of depositors in bank runs (e.g., Diamond and Dybvig, 1983) may be explained by such models.
However, in many real world situations both informational externalities as well as payo¤ externalities seem to be present at the same time. For example, the choice of software or hardware is often described as a situation with network externalities.3 While one’s choice of such a product may very well convey information about its quality to later potential users, it is also the case that the more users adopt the same system, the easier becomes interaction with them. For such positive payo¤ externalities to materialize it does not
1
Aninformation cascadeis said to occur when it becomes rational to ignore one’s own private information and instead follow one’s predecessors’decisions. Since no further information is revealed once an information cascade has started, ine¢ ciencies occur even though each individual is behaving rationally.
2For surveys of this literature see e.g., Bikhchandani, Hirshleifer, and Welch (1998) and Gale (1996), for a generalization see Smith and Sorensen (2000), and …nally for a recent textbook treatment see Chamley (2004).
3
See e.g., Katz and Shapiro (1985), or Church and Gandal (1992). Based on internal documents made public in the antitrust trial, Bresnahan (2004) provides an overview of network e¤ects in the software industry as perceived by Microsoft.
matter (much) whether other users have already adopted the system or whether they will do so soon. Timing may be important, however, in other situations. Consider the choice of a research area. If one aims at maximizing the number of citations, one should enter a new research area as citations can obviously only be directed at older papers. A new research area may turn out to be a dead end, though. Negative payo¤ externalities may be caused by overcrowding (e.g., in restaurants, supermarket check–out counters, parking lots, etc., where one’s utility decreases with the number of predecessors who chose the same restaurant, cashier, or parking lot but where one is not bothered by people arriving later). Finally, there are situations where one is punished for taking the same action as a predecessor but is rewarded for successors. Avant-gardist and fashion leaders fall into this category as well as the unlucky participants in snowball systems or chain letters.
The purpose of the current paper is threefold. First, the paper presents a broad– scaled replication of existing laboratory studies on information cascade models. Second, the paper aims to extend this literature by experimentally studying various settings in which additionally payo¤ externalities are present. And third, the paper investigates the importance of reputation for cascade behavior, or in other words, the in‡uence of role models.
With respect to replication, our experiment di¤ers from earlier cascade experiments in that (1) we replicate those experiments with more than 6000 subjects, many more than usual. (2) The large number of subjects allows us to test a number of variations that may potentially be important (e.g., longer sequences of decisions). (3) Instead of the usual undergraduate student population, we use a diverse subject pool, including 267 consultants from an international consulting …rm. More than 40% of our subjects hold a Ph.D. or are currently enrolled in a Ph.D. program. A majority of subjects has a background in the natural sciences. (4) Finally, we deviate from the usual laboratory setting by utilizing the
internet for our experiment.4
The second purpose of our paper is to study the e¤ects of payo¤ externalities. The bulk of previous experimental research has focused on settings without payo¤ externali-ties. However, as argued above, many, if not most, real world examples of herding have a payo¤ externality component. As payo¤ externalities may come in various forms, we consider positive payo¤ externalities (which should reinforce herding) as well as negative externalities (which should slow down herding). Since we assume sequential decision mak-ing, we will further di¤erentiate between externalities that apply only to predecessors, only to followers, or to both (but possibly in di¤erent ways).
Finally, our paper aims to study the e¤ect of subjects’reputation on the (herd) behav-ior of later decision-makers. We do this by informing subjects of the cumulative payo¤s their predecessors have achieved in earlier unrelated rounds. We hypothesize that subjects are primarily in‡uenced by the subject with the highest earlier payo¤ (i.e., the highest rep-utation), despite the fact that this information is irrelevant in a rational Bayesian model.
Related literature There is by now a large (mostly theoretical) literature on network
e¤ects.5 Surprisingly, there is, however, only a rather small theoretical literature on the interplay between information cascades and network e¤ects, and the few papers that do exist di¤er in some important aspects from Bikhchandani, Hirshleifer, and Welch (1992) (see e.g., Choi, 1997; Vergari, 2004; Frisell, 2003; Jeitschko and Taylor, 2001; Corsetti,
4
Arguably, for many people who buy and sell goods on the internet, use internet banking and brokerage services etc., the internet is probably by now a very natural setting for decision making. Nevertheless, conducting experiments on the internet is still novel. For experiments that have been conducted over the internet, see e.g., Forsythe et al. (1992, 1999), Lucking-Reiley (1999), Anderhub, Müller, and Schmidt (2001), Charness, Haruvy, and Sonsino (2001), Shavit, Sonsino, and Benzion (2001), Bosch-Domenech, Montalvo, Nagel, and Satorra (2002), and Güth, Schmidt, and Sutter (2003). For technical issues, see e.g., Greiner, Jacobsen, and Schmidt (2002). The internet has also been used to provide a platform to run economic experiments for interactive learning (see e.g., Holt, 2002).
5For a recent survey of this literature, see e.g., Farrell and Klemperer (2004). For a textbook treatment, see e.g., Shy (2001).
Dasgupta, Morris, and Shin, 2004; Dasgupta, 2000).6 A combination of Bikhchandani,
Hirshleifer, and Welch’s (1992) seminal model with payo¤ externalities does not seem to have been treated theoretically. Given the relative scarcity of theoretical work on the interplay between information cascades and network e¤ects it is not surprising that there is also almost no experimental work in this area.7 The only experimental paper introducing payo¤ externalities in the Bikhchandani, Hirshleifer, and Welch (1992) framework we are aware of is Hung and Plott (2001).8 They study treatments in which subjects are rewarded if a majority of decisions was correct or if the subject’s action agreed with the majority, respectively. The externalities in our experiment are, however, of a di¤erent form.
The remainder of the paper is structured as follows. In Section 2 we describe the basic experimental settings. In Section 3 we derive the theoretical predictions for the various treatments. While we …nd a multitude of equilibria in a treatment with network e¤ects, there do not seem to exist pure strategy equilibria in treatments where only the followers cause positive payo¤ externalities. In Section 4 we describe the experimental procedures in detail. Since internet experiments are still relatively novel, we explain how we resolved the issues of recruitment, payment of subjects, and the implementation on the internet.
Section 5 contains our results. Section 5.1 deals with replication of earlier cascade experiments. Besides presenting our own results, we review and compare results from 12
6In Choi’s (1997) model herding is not driven by private information but by the interplay of risk aversion and network e¤ects (see also Vergari, 2004). Frisell (2003) considers a waiting game, where two …rms with private information regarding the most pro…table niche have to decide about entry into a market with horizontally di¤erentiated products. While Jeitschko and Taylor (2001) study an investment game where randomly matched agents play pairwise coordination games, Corsetti, Dasgupta, Morris, and Shin (2004) explore the in‡uence of a large trader in a model of speculative currency attacks with private information. Finally, Dasgupta (2000) studies a model relatively close in structure to Bikhchandani, Hirshleifer, and Welch (1992). However, he considers continuous signals and a network externality of a relatively extreme form: agents may realize a positive pro…t only if all agents coordinate on the same action.
7Beginning with Anderson and Holt (1997), there is by now a large experimental literature on information cascades in the absence of payo¤ externalities. This literature will be (partly) reviewed in Section 5.1 below. 8Guarino, Huck, and Jeitschko (2003) provide an experimental test of the above mentioned paper by Jeitschko and Taylor (2001). See also Schotter and Yorulmazer’s (2004) experimental study of bank runs, where both asymmetric information and (negative) payo¤ externalities are present.
earlier experiments that are scattered in the literature. Compared to these studies we …nd that subjects behave less frequently in line with theory. We attribute this observation to the fact that we consider longer decision sequences as well as asymmetric priors (i.e., one of the alternatives is more likely to be successful from an ex-ante perspective). Separately considering various subgroups of our diverse subject pool reveals that there are no signi…-cant di¤erences across educational background or sex of the subjects, but that consultants have a somewhat larger tendency to follow their own signal. Section 5.2 presents the re-sults from the treatments with payo¤ externalities. We study uniformity, volatility, and predictability of behavior in these treatments. We …nd that subjects seem to behave my-opically (i.e., they tend to take only the decisions of their respective predecessors, but not the behavior of their successors, into account).9 Finally, Section 5.3 deals with reputation e¤ects in the basic Bikhchandani et al. model. While from a theoretical point of view reputation should not matter, we …nd that subjects’behavior is signi…cantly in‡uenced by the behavior of the predecessor with the highest reputation, and these “success models” were on average indeed more likely to pick the successful alternative. Section 6 concludes. Instructions of the experiment are contained in an Appendix.
2
The Experiment
In the experiment subjects had to choose sequentially between two “investment opportu-nities”A andB. Only one of the two could be successful and, if so, would pay 10 “Lotto– Euros”. The unsuccessful alternative paid nothing. Subjects were told the a priori prob-ability that investment A was successful, P(A) = 0:55 (and consequently, P(B) = 0:45). Furthermore, they were told that they would receive a tip by an investment banker that was reliable with probabilityP(ajA) =P(bjB) = 0:6. Sessions with these probabilities are
9
For related empirical evidence on the apparent lack of forward-looking behavior in the presence of payo¤ externalities see Tucker (2004).
denoted by 55-60. In some treatments we conducted additional sessions with the probabil-ity combination 50-66, which is the one most often used in the literature (see e.g., Anderson and Holt, 1997).10
Subjects were informed that all prior subjects in their group had received a tip by other investment bankers and that these tips were independent of theirs (see the Appendix for a translation of the instructions). Subjects were able to observe the decisions of their predecessors but, in general, not their signals.
We consider two principal versions of this model.11 In a …rst version a subject’s payo¤
depends exclusively on his own decision. This version is equivalent to the basic model studied by Bikhchandani, Hirshleifer, and Welch (1992) and is denoted byBHW. For com-parison, we also include a treatmentBHW+AS in which additional to predecessors’actions also all their signals were observable. There is also a “reputation” treatment BHW+R,
which will be discussed in more detail in Section 5.3.
In a second version of the above model we introduce four di¤erent forms of payo¤ externalities, i.e., we consider treatments in which payo¤s also depend directly (positively or negatively) on the decisions of others. Table 1 lists the main features of all treatments. In treatment Network subjects receive an amount x for each other subject in their group that chooses the same action. This payo¤ structure is supposed to capture network externalities. Examples are the choice of software or mobile phone operators. There, it is not the quality of the product alone on which a choice should be based. As the utility from such products is increasing in the number of adopters, it is also important which product is selected by the majority of other consumers.
In treatmentFollower subjects receive an amountxonly for those subjects that decide
1 0
The probability combination 50-66 has, however, the disadvantage of requiring a tie–breaking assump-tion in many cases.
1 1
In a companion paper we focus on treatments with market prices for the investment opportunitiesA
and B (as in Avery and Zemsky, 1998). Those treatments were conducted in the same experiment (see Drehmann, Oechssler, and Roider, 2004).
Table 1: Treatments
treatment description # of groups
BHW Bikhchandani/Hirshleifer/Welch 63/12/15
BHW+AS BHW + signals of all predecessors observable 70/12/9
BHW+R BHW + cumulative payo¤s of predecessors observable (reputation)
29
Network BHW + receive xfor each group member who chooses same alternative
12/6
Follower BHW + receive xfor each follower who chooses same alternative
26
Early bird BHW + payx for each predecessor who chose same alternative
26
Hipster BHW + pay (receive)x for each predecessor (follower) who chose same alternative
12/6
Note: x/y/z denotes x groups with probability combination 55-60, y groups with 50-66, and z groups with consultants (also 55-60). In treatmentsBHW+R,Follower andEarly bird the probability combination is 55-60; in treatmentsNetwork andHipster xis either 0.4 or 1; denotes that there were 12 groups withx= 0:4and 6 withx= 1; in treatmentsFollower andEarly bird xis always 0.4.
later in their group and choose the same action. Examples for such one–sided network externalities are choices on software that is only upwards compatible or the choice of a research topic by a scientist who is concerned about the number of citations to his work. Clearly there can be no citations from papers that have already been published.
In treatmentEarly bird subjects have to payxfor each predecessor who chose the same action as they. This kind of payo¤ externality is typical for situations where overcrowding is an issue as in restaurants, movie theaters, beaches, etc.
Finally, treatment Hipster is a combination of Follower and Early bird as subjects receive x for each follower who chooses the same action but have to pay x for each pre-decessor with the same action. Examples include fashion leaders, avant-gardist, and the participants in snowball systems or chain letters.
3
Theoretical Predictions
As mentioned above, there does not seem to exist in the literature a theoretical treatment of a model combining the Bikhchandani, Hirshleifer, and Welch (1992) model with payo¤ externalities. When subjects’ payo¤s are either independent of others or depend only on the behavior of predecessors, (unique) equilibrium predictions are straightforward to obtain by backward induction. This is the case in treatmentsBHW andEarly bird. At …rst sight it may be surprising that for the other treatments either there exist multiple pure–strategy equilibria or none seem to exist at all. However, it is well known from the literature on network e¤ects that this may happen as the strategic situation may well resemble those of coordination or mis–coordination games. This feature lights up in treatmentsNetwork,
Follower, and Hipster.
Table 2 presents a non–exhaustive list of candidates for pure strategy (perfect Bayesian) equilibria given probability combination 55-60.
Table 2: Candidate equilibria for probability combination 55-60
candidate …rst player’s strategy strategies of players 2 through 20
bhw follow own signal A if 1;B if 2; otherwise follow own signal
uniform follow own signal follow action of player 1
reverse follow own signal choose opposite of player 1
stubborn choose A choose A
Note: denotes the net number ofasignals (#asignals #bsignals) that can be imputed from the actions of predecessors; in treatmentBHW+AS denotes the net number of directly observed
asignals.
In treatments BHW and BHW+AS there is a unique perfect Bayesian equilibrium, which depends in a simple way on the net number of signals that can be imputed from the actions of predecessors and the own signal (for details see Bikhchandani, Hirshleifer, and
Welch, 1992, or Drehmann, Oechssler, and Roider, 2004). We call this thebhw equilibrium.
It can easily be checked that this equilibrium does not exist for any of the treatments with payo¤ externalities.
Since in treatment Early bird payo¤s depend only on the actions of predecessors and the own action, the game can again be solved by backward induction. It turns out that cascades happen but they are endogenously broken once su¢ ciently many predecessors have chosen the same action. From this point on, actions may reveal signals again, which may, in turn, lead to a new cascade. In comparison to theBHW treatment, where cascades once started last until the end of the group, we should see shorter cascades in treatment
Early bird.
Treatment Network allows for a multiplicity of equilibria. All of the candidates uni-form, reverse, and stubborn can be supported as perfect Bayesian equilibria with suitably chosen o¤–equilibrium beliefs.12 More complex equilibria, in which players 2 through 20 act di¤erently, also exist. Finally, in treatmentsFollower and Hipster none of the candidates listed in Table 2 are Nash equilibria, and we conjecture that no pure strategy equilibria exist. As an example consider the uniform equilibrium candidate in treatment Follower. If the …rst players receives and follows a b signal, all subsequent players are supposed to playB. However, the last player, who does not have any followers, wants to deviate if he receives an a signal because his a signal and the …rst player’s b signal cancel and we are back to the a priori probability, which with 0.55 is in favor ofA.
Finally, given the potential complexity of the equilibrium strategies in some of the payo¤-externality treatments, there might be a tendency for players to behave myopically (i.e., to ignore the behavior of their respective successors altogether). If subjects indeed behave myopically, it turns out that treatment Follower yields the same prediction as
BHW. Likewise,Early bird and Hipster become indistinguishable from each other. Below
we shall also test this behavioral assumption, which is not uncommon in the literature on network e¤ects.
4
Experimental Design
More than 6000 subjects participated in our internet experiment, which was available for a period of about six weeks in the spring of 2002 on our web sitehttp://www.A-oder-B.de, which is German for A-or-B. Subjects decided in sequence and were able to observe the actual decisions of prior participants in their respective groups. In general, the group size was 20.13 Subjects were asked to make decisions in three independent groups, thus in total there were more than 18000 decisions. We call the …rst decision stage 1, the second stage 2, etc. The last column of Table 1 lists the number of groups that participated in our experiment, separately for each combination of treatments, probabilities, and whether subjects came from the general subject pool or the control experiment with the consultants. Payo¤s in “Lotto–Euros” were calculated as follows. If a subject chose the correct investment, he received 10 Lotto–Euros. This was the …nal payo¤ for this task in theBHW
treatments. In the treatments with payo¤ externalities, once all subjects in the respective group had decided, the payo¤s of the subjects were raised or lowered by the amount of the respective payo¤ externalities. In treatments with negative externalities (Early bird and
Hipster) subjects additionally received an endowment of 5 (ifx= 0:4) or 10 (ifx= 1) for each task to avoid losses because negative payments are obviously very di¢ cult to enforce in any experiment, let alone an internet experiment.
1 3
Except in two cases: in the general subject pool, the group length in treatmentBHW+AS was 10; with the consultants, the average group length in treatmentBHW (BHW+AS) was 7 (8).
4.1 Recruiting and Payment
The experiment was announced in several ads in the science section of the largest Ger-man weakly newspaperDie Zeit, two popular science magazines, and two national student magazines. Posters were distributed at most sciences faculties at German universities. Finally, emails were sent to Ph.D. students and postdocs in science and economics depart-ments at 35 universities in Germany. The web site www.A-oder-B.de was linked to the Laboratory for Experimental Research in Economics at the University of Bonn and to the sponsor McKinsey & Company to demonstrate that the experiment had a proper scienti…c background and that the promised …nancial rewards were credible.
All payo¤s in the experiment were denoted in “Lotto–Euro”. Each Lotto–Euro was a ticket in a lottery to win one of our main prizes. In total there were 11 prizes of 1000 Euro each. Importantly, the odds in those lotteries were …xed in advance and known to subjects: each subject, when logging in on our website was told explicitly the odds per lottery ticket for winning one of our main prizes. Thus, maximizing the probability of winning one of the prizes was equivalent to maximizing the number of lottery tickets. All winners were noti…ed by mail, and their prize money was paid through bank transfers.
In a Phase I of the experiment, 1409 subjects played with high powered incentives, where each of 40000 lottery tickets had an equal chance of winning one of 5 prizes of 1000 Euros. Since subjects played on average for about 15 minutes, they were making an expected hourly “wage” of 14.19 Euros, which is comparable to a very good student job and to pay in laboratory experiments. In a Phase II, each of 90000 lottery tickets had an equal chance of winning one of another 5 prizes of 1000 Euros. Finally, in a Phase III, 1162 subjects competed for the remaining 1000 Euros. Only in this Phase III of the experiment (where almost no monetary incentives were provided) subjects did not know how many lottery tickets were issued in the respective phase. This payment scheme was due to the
fact that an unexpected large number of subjects participated in our experiment. But it also gives us the chance to test the e¤ects of monetary incentives on behavior in such a setting.
Additionally, there was a control group of 267 consultants from McKinsey & Company, an international consulting …rm, who participated in the experiment on the same web site a couple of weeks before the start of the actual experiment. The subjects of the control experiment were recruited by an internal email to all German McKinsey consultants. Sub-jects knew that all other subSub-jects were also consultants. About a third of those addressed participated. These subjects had the chance to win 8 vouchers for a nice dinner for two in a restaurant each worth 150 Euros.
4.2 Subject Pool
In total, 6099 subjects …nished our experiment of which 5832 subjects participated in the main experiment and 267 in the control experiment with consultants.14 Table 3 lists some of the main characteristics of the combined subject pool (including the control experiment with consultants).
In contrast to most experiments in economics, our subjects come from a broad range of …elds. Figure 1 shows the frequencies of the main subject groups. Each bar in Figure 1 shows the number of subjects who study for or have …nished a …rst degree, the number of subjects who currently are Ph.D. students, and the number of subjects who have …nished a Ph.D.15 Considering the number of Ph.D. students and Ph.D.’s we believe we succeeded in recruiting a fairly bright subject pool.
1 4788 individuals logged on but did not …nish the experiment. Their decisions were not included in the historyHtsince they did not face monetary incentives (payment was conditional on …nishing all three stages of the experiment).
1 5Given that each time that we sent out emails to Ph.D. students and post-docs to advertise the experi-ment, there was immediately a peak in access to our webpage, one can be con…dent in these numbers.
Law Li ber al A rts Medi cine Engi neer ing M isc. Econom ics Scienc es N u mb er o f s ub jec ts 2500 2000 1500 1000 500 0 Ph.D completed Ph.D student Student or first degree completed
Figure 1: Composition of the subject pool. (Note: “Sciences” includes physics, chemistry, mathematics, and computer science; “Economics”includes economics, business administration, and related subjects; “Medicine” includes medicine, psychology, and dentistry; “Liberal Arts” includes all languages, history, and pedagogy. “Misc.” stands for miscellaneous …elds.)
4.3 Implementation
When arriving on our web site, subjects read a screen that introduced the general problem and the rules of the game. Subsequently, subjects were asked for some personal information (name, mailing address, email, …eld of study, age, etc.), and subjects were only allowed to play if all information requested was actually provided. This was also a measure to prevent subjects from playing twice: in order to win in the lottery, one had to give a correct mailing address, and the program ensured that the same name-postal code combination as well as the same email address could only play once. We also used cookies to prevent using the
Table 3: Properties of the subject pool
Average age 28.3
% of female subjects 27.8
% completed (at least) …rst university degree 56.9
% current students 36.4
% non–students 6.7
% completed Ph.D. 13.7
% current Ph.D. students 31.3
same computer twice.16
After entering the personal information, subjects were randomly placed in a currently active group,17 and had to make their …rst decision. Afterwards they were randomly placed in another active group for the second task and then in a third group for the …nal task. No feedback about results was given until the subject had completed all three tasks, and even then they were only told how many ”Lotto–Euro” they had won. Usually the tasks for each subject came from di¤erent treatments. Finally, we asked subjects for voluntary feedback as to how they formed their decision, and 687 subjects sent response emails.
5
Results
5.1 Replication
To make our results comparable to earlier experimental studies we shall concentrate on the following three measures. (1) Average rationality under common knowledge of rationality (ruck), which is de…ned as the fraction of subjects who behaved according to a Perfect Bayesian equilibrium under the assumption that all predecessors are commonly known to
1 6
It will never be possible to completely prevent clever people from playing more than once. However, we are con…dent that not many of such attempts were successful, and, given the size of the subject pool, those few probably do not matter much.
1 7A group wasactive when it was neither full nor closed (i.e., when another subject was active in this group). We also ensured that subjects who logged on at about the same time were allocated to di¤erent treatments to prevent “observational learning” in case two subjects sat next to each other in a computer pool.
be Bayesians.18 (2) The fraction of cases in which subjects rationally decided against their
own signal if they are in a cascade is denoted by casc. Arguably, casc is a harder test for cascade theories sinceruck includes all the cases in which subjects (rationally) follow their own signal. (3) The fraction of cases in which subjects followed their own signal is denoted by own. For comparison, we also report the equilibrium value of own, denoted by own
that would have obtained had all subject behaved according to ruck (as de…ned above).
Table 4: BHW treatments
subject pool treatment prob. all subjects subjects on pot. eq. path comb. ruck casc own own ruck casc own own
general BHW 55-60 .66 .34 .75 .59 .86 .74 .74 .90 BHW+AS 55-60 .72 .41 .74 .68 - - - -consultants BHW 55-60 .68 .16 .85 .66 .90 1.00 .83 .93 BHW+AS 55-60 .78 .52 .69 .60 - - - -general BHW 50-66 .78 .45 .75 .62 .95 .78 .84 .77 BHW+AS 50-66 .76 .59 .69 .69 - - -
-Note: The average length of potential equilibrium paths is 6 in treatmentBHW 50-66 and 3 in the two remaining cases; as in treatmentBHW+AS signals of predecessors were public information, we do not di¤erentiate in this case whether or not a subject observed a potential equilibrium history of decisions.
Table 4 lists those measure for our BHW andBHW+AS treatments, and for our sub-sample with consultants (who also played treatmentsBHW and BHW+AS). To construct Table 4 we have pooled data from all phases (recall that di¤erent incentives were provided in di¤erent phases) and all stages (whether a task was …rst, second or third) as it turned out that neither the phase of the experiment nor the stage of the task had a signi…cant in-‡uence on those results according to MWU-tests. Table 4 lists the above de…ned measures for all subjects and for those that are on apotential equilibrium path. We say that subjects
1 8For a decision that, given the history of imputed signals, obviously violates Perfect Bayesian equilibrium, we let players suppose that the deviator followed his private signal. Dominitz and Hung (2004), who privately elicit the beliefs of all subjects after all decisions, report that subjects indeed seem to act on this assumption.
are on a potential equilibrium path as long as there is no prior decision that obviously violates behavior in a Perfect Bayesian equilibrium from the viewpoint of a player who cannot observe the private signals of predecessors.
First, a look at all subjects shows that from a theoretical perspective subjects rely too heavily on their own private signal asown is weakly aboveown in all cases. As a result, in our main treatment,BHW 55-66, subjects act in accordance with theory in only 66% of cases.19 Even more dramatic is the picture with respect tocasc. Only in 34% of cases did subjects decide against their signal but in accordance with Bayesian updating.20 Those
numbers are lower than those found previously in most of the literature. In the following, we provide a brief overview of earlier experiments on information cascades and o¤er some explanations for the interesting observed behavioral di¤erences.
Table 5 lists the results of all cascade experiments implementing the basic setup of BHW that we were able …nd in the literature.21 While the experiments di¤er with respect to a number of design issues, most notable the number of players in a sequence and the probability combinations, most values of ruck and casc are roughly comparable and are higher than those in our experiment.
What could account for those di¤erences? One possible explanation may be that de-cisions are more di¢ cult on average when 20 subjects decide in sequence rather than the usual 6.22 To test for this we look at the decisions of our …rst 6 subjects in each group.
1 9Note that after a subject had made his decisions, we also asked for his prediction regarding the prob-ability of A being successful (this prediction was, however, not renumerated). Interestingly, despite the relatively low value ofruck, like Dominitz and Hung (2004) we …nd that in over 90% of cases subjects chose actions that were consistent with their probability judgement.
2 0
Recently, Kuebler and Weizsaecker (2004) have provided some evidence that longer cascades tend to be more stable indicating that, counter to theory, subjects perceive decisions in a cascade to be informative. They consider subjects already in a cascade and sort them according to the number of their cascade-predecessors. In …ve out of the six studies they review, averagecasc is lower in the …rst half of observation than in the second half. This is also the case in our experiment, albeit to a smaller extent.
2 1
We thank Lisa Anderson and Charlie Holt for kindly providing their data.
2 2Huck and Oechssler (2001) show that ruck values are substantially lower when decisions are more complex.
Table 5: Previous BHW cascade experiments
study treatment prob. group size ruck casc
Alevy et al. (2003) symmetric, students 50-66 5 .95 .89 Anderson/Holt (1997) symm., no public sig. 50-66 6 .92 .73
Anderson (2001) 2$ 50-66 6 .70
Cipriani/Guarino (2004) …xed price 50-70 12 .83 Dominitz/Hung (2004) "replication" treatment 50-66 10 .88
Goeree et al. (2004) 50-66 20 .64
Hung/Plott (2001) individualistic 50-66 10 .77 Kübler/Weizsäcker (2004) NC 50-66 6 .78 Oberhammer/Stiehler (2001) 50-60 6 .86 .73 Stiehler (2003) only equilibrium histories 50-60 6 .97 .93 Willinger/Ziegelmeyer (1998) treatment 1 50-60 6 .64 Ziegelmeyer et al. (2002) blue line, exp. 1&2 55-66 9 .69 Note: Only studies that implement the BHW model are included. In some cases, values forruck orcasccould
not be determined from the information given in the respective papers. Also, in some cases it was unclear whether all observations were counted or only those on a potential equilibrium path. The probability combination (prob.) is given as x-y, which denotes an a priori probability forA ofx%and a signal precision ofy%.
And indeed, for the …rst 6 subjects ruck is 82%, which is closer to the numbers found in the literature. Additional support for this hypothesis is provided by the results of Goeree, Palfrey, Rogers, and McKelvey (2004). They also consider sequences of 20 subjects and report one of the lowest values for casc (see Table 5). Another aspect emerges when we consider subjects on a potential equilibrium path (right panel of Table 4). On average, po-tential equilibrium paths have length 6 (which again coincides with the length of sequences considered in many of the earlier studies). On those paths, values of ruck and casc are very high (and comparable to those reported in Table 5), which indicates that subjects become confused as soon as they observe deviations from a potential equilibrium path. Interestingly, on potential equilibrium paths, subjects do rely less often on their private information than predicted by theory under probability combination 55-60.
A second possible explanation is that subjects simply mistrust the behavior of their predecessors on the internet more and consequently rely more readily on their own signals. However, this consideration should not matter in treatment BHW+AS where all signals of predecessors were observable and the payo¤-maximizing decision is simply a matter of forming conditional expectations. Yet, the measures ruck,own, and casc are not substan-tially higher (even though for probability combination 55-60 both for the general subject pool and the consultants ruck and casc are signi…cantly higher in BHW+AS at the 1% level according to MWU–tests). Also, in the control experiment with consultants, where all participants had a relatively good idea about the types of their predecessors, the reliance on the own signal is even more pronounced,23 and the number for casc is substantially lower compared to the general subject pool. It seems that consultants are more reluctant to rely on the decisions of others. Interestingly, Alevy, Haigh, and List (2003) also …nd in their experiment that professional traders have lowerruck and casc values than college
2 3
This cannot be explained by a higherown in equilibrium. For consultants, given the random draw of signals, equilibriumown would have been 0.66 whereas forBHW 55-66 it would have been 0.59.
students.
A third possible explanation is that the probability combination 55-60 (with asymmetric prior) is more di¢ cult than the (symmetric) combination 50-66, which was often used in the literature. For example, a subjects with absignal on the second position should already ignore his signal if the …rst subject choseA for 55-60 but not for 50-66. This hypothesis is supported by the signi…cantly higher numbers for ruck (78%) andcasc (45%) in 50-66.24
Finally, the level of payo¤s may play a decisive role in complex decision problems. For example, Anderson (2001) shows that errors decrease substantially when the payo¤ for a correct decision is increased from 0 to 2$. In our experiment subjects earned about 1.25 Euros for a correct decision in Phase I and 0.55 Euros in Phase II.25 While we do not observe a signi…cant di¤erence in ruck between Phases I and II, it is possible that the lower payo¤s in combination with the …rst and the third explanation above are responsible for the values observed in Table 4.
It is also interesting to test whether di¤erent subject characteristics in‡uence ruck and
casc. However, we do not …nd signi…cant di¤erences between subjects holding a Ph.D., Ph.D. students, or others, and between male and female subjects. In treatmentBHW the McKinsey consultants di¤er from the general subject pool by showing signi…cantly higher values for own and lower values for casc (at the 1% respectively 5% level according to MWU-tests).
5.2 Payo¤ Externalities
Figure 2 presents a …rst view on how the various payo¤ externalities in‡uence behavior.
We call the di¤erence between the number ofAandB decisions thedecision imbalance, and Figure 2 depicts the distribution of decision imbalances after the last player. This
2 4With respect toruck (casc) the di¤erence is signi…cant at the 1% (10%) level. 2 5
0% 20% 40% 60%
BHW Early bird Follower
Hipster Net work
-20,00 -10,00 0,00 10, 00 20, 00 0% 20% 40% 60% -20,00 -10,00 0,00 10, 00 20, 00
Figure 2: Distribution of decision imbalances after the last period (pooled over x).
variable captures how many subjects in a given group make the same choice, and hence it measures uniformity.26 TreatmentNetwork (where a subject’s payo¤ is the larger, the more players make the same choice) clearly stands out as the only treatment in which extreme imbalances occur. That is, in this treatment subjects often coordinate on the same choice, and, as we will see below, observational learning early in the process is frequently pivotal for the outcome. On the other hand, treatmentsEarly bird andHipster (where one’s payo¤ is the lower, the more players have made the same choice in the past) produce very balanced distributions centered around 0: the decision imbalance is exactly 0 in 26.9% of cases in
Early bird and in 61.1% of cases in Hipster, whereas the same holds only in 9.2% of cases in BHW, in 7.7% in Follower, and in 0% in Network. Kolmogorov–Smirnov tests reveal that the distribution forNetwork is signi…cantly di¤erent from all other treatments except
2 6
Follower (at the 5% level or better). Recall that if subjects are myopic, there should not be any di¤erence between Early bird and Hipster, and between BHW and Follower. And indeed, there are no signi…cant di¤erences in the distributions of decision imbalances for those two pairs. In all other pair-wise comparison one cannot reject that the distributions are signi…cantly di¤erent (at the 5% level or better).
Table 6: Number and length of runs
treatment x number of runs average length Aruns average length B runs
Network 0.4 8.55 3.51 1.39 Network 1 4.50 4.38 4.50 Follower 0.4 9.58 2.62 1.53 Early bird 0.4 10.38 2.14 1.70 Hipster 0.4 11.92 1.79 1.56 Hipster 1 12.33 1.68 1.57 BHW - 9.84 2.45 1.59
Note: Probability combination 55-60; general subject pool.
A second indicator of possible behavioral di¤erences in the payo¤-externality treatments is the number and length ofruns in the data. A run is a sequence of consecutive subjects who made the same decision. Hence, the number of runs captures whether a group exhibits relative stability or whether it is characterized by a rapid succession of short-lived fads. If there are positive payo¤ externalities (as in Network) we would expect longer (and therefore fewer) runs, i.e., runs should not be as fragile. When it is harmful to have many predecessors who made the same choice, runs should be shorter and more frequent. Table 6 lists the average number and length of runs per group for our treatments (separate forA
andB runs).27 As expected,Network has the lowest number and highest average length of runs. Again in accordance with myopia,BHW and Follower seem to show runs of similar (medium) length and frequency. The shortest and most frequent runs are found forEarly
bird and Hipster. The fact thatB runs are shorter on average in all but one case might be explained by the higher a priori probability forA.
A third interesting aspect of the data is predictability. For example, is it possible to predict early on which product will capture a larger slice of a market? Above it has already become clear that in treatment Hipster an equal split is very likely: there, in 61.1% of cases the decision imbalance after the last player is exactly zero.28 Indeed, in this treatment decision imbalances do in general not move too far away from zero: looking at decision imbalances pooled over all players (not just the last one) reveals thatHipster
produces both the lowest mean (:84) and the lowest standard deviation (1:78) across all treatments.
In order to uncover potential predictability in the remaining treatments, we ask whether one can forecast the majority decision in a group after observing the …rstnplayers. Table 7 shows correlations between the sign of the decision imbalance after playern= 2;5;10;15
and the sign of the decision imbalance after player 20. Note that a decision imbalance is positive if a majority of subjects chose A; and vice versa. Treatment Network with
x= 1 shows the highest predictability. Already after the second player the correlation is 0.86 and signi…cant at the 5% level. Follower and Network with x = 0:4 also show high correlations. Given that subjects inHipster frequently split 50:50, it is not surprising that in this treatment it is hard to predict which alternative is chosen (slightly) more often.
Given the multiplicity of equilibria for treatment Network for x= 0:4 it is interesting which, if any, of those equilibria can be observed in the data. We classify the decisions of a group of 20 subjects as in accordance with an equilibrium if at most 4 subjects deviate from the equilibrium path. In this sense, of the 12 groups with x = 0:4, 6 groups can be classi…ed as one of the equilibria listed in Table 2, namely, 2 asstubborn, and 4 asuniform
2 8
Table 7: Predictability of majority choice
correlation between the sign of the decision imbalance after player 20 and after player...
treatment x 2 5 10 15 Network 0.4 0:28 0:52 0:67 1:0 Network 1 0:86 0:93 0:93 0:93 Follower 0.4 0:21 0:71 0:91 0:80 Early bird 0.4 0:20 0:32 0:71 0:65 Hipster 0.4 0:30 0:71 0:13 0:71 Hipster 1 1:0 0:32 0:32 0:45 BHW - 0:08 0:23 0:61 0:75
Note: Probability combination 55-60; general subject pool; signi…cant at 1%-level; signi…cant at 5%-level; signi…cant at 10%-level.
orstubborn.29
5.3 Reputation E¤ects
TreatmentBHW+R is identical to treatment BHW except that subjects were able to ob-serve not only the actions of their predecessors but also the cumulative payo¤s (denoted in "Lotto-Euros") from the two decisions those subjects made in the …rst two (unrelated) stages of the experiment.30 In a rational Bayesian model, this extra information is ir-relevant. However, we suspected that subjects would rely more on the decisions of the predecessors with the highest payo¤s (the “success models”). That is, subjects with higher payo¤s have a better reputation and are imitated more often.
Figure 3 shows that subjects are indeed in‡uenced by the decision of the predecessor with the highest reputation (i.e., the highest cumulative payo¤ in stages 1 and 2). Re-gardless of the own signal, an A decision by this predecessor signi…cantly increases the
2 9If the …rst subjects receives anasignal, the two equilibria are indistinguishable. 3 0
Recall that each subjects had to make three decisions (stage 1 through 3). BHW+Rwas always played on stage 3.
Own signal A B 1,0 ,8 ,6 ,4 ,2 0,0 Success model chooses B chooses A
Figure 3: Fraction of subjects choosingA depending on the choice of the predecessor with the highest reputation and on the own signal. (Note: Including only subjects that had at least one predecessor and including only cases where the predecessor with the highest reputation was unique (547 out of 551 cases)).
frequency of the choice ofA according to MWU–tests (at the 5% level).31
>From an ex-post perspective, did it make sense for subjects to follow the respective success model? While on average subjects chose the successful alternative in 56% of cases, success models did so in 62% of cases. Hence, these subjects were indeed (somewhat) more successful in picking the right alternative, and imitating their behavior made sense.32
3 1This result is supported by a logit regression. Even when variables like a subject’s signal and the signal imbalance are included, the decision of the predecessor with the highest payo¤ has a signi…cant in‡uence on the decision to choose a certain action.
3 2
In a similar spirit, based on theoretical work by Celen and Kariv (2004), Celen, Kariv, and Schotter (2003) experimentally study the role of advice in an information cascade model. While in theory advice and actions are equally informative, Celen, Kariv, and Schotter (2003) …nd that (i) subjects appear to be more willing to follow their predecessors’advice than their predecessors’actions, and that (ii) the presence of advice is welfare-enhancing.
6
Conclusion
In a large–scale internet experiment we investigated information cascade models with and without payo¤ externalities. Reassuringly, while our subject pool is quite diverse (with large fractions of subjects having a background in the natural sciences and holding or studying for a Ph.D.), various subgroups of subjects do not seem to behave signi…cantly di¤erent with respect to the main research questions.
For the base treatment without payo¤ externalities, compared to earlier results in the literature we …nd a substantially lower percentage of subjects who behaved according to theory. We explain this deviation through a combination of the probability combination (asymmetric prior vs. symmetric prior), the number of subjects deciding in sequence (20 vs. 6), and the level of payo¤s. While our results do not question the fact that information cascades do happen in experiments, they certainly show that cascades – depending on the setting – may be rarer and shorter than predicted by theory and suggested by earlier experiments.
Surprisingly, there is only a very small literature on the interplay between information cascades and payo¤ externalities, either theoretical or experimental (see e.g., Hung and Plott, 2001). We studied several di¤erent forms of payo¤ externalities, positive and negative ones and those that apply to all group members or only to predecessors or followers. The experimental results are by and large compatible with the theoretical predictions. With positive externalities (network e¤ects) cascades become longer and more robust, whereas with negative externalities they become shorter and more fragile. In most cases we could not reject the hypothesis that subjects behaved myopically as treatments that have the same theoretical solution under myopia yield very similar results. The form of payo¤ externality was also found to have strong e¤ects on the predictability of the majority decision. With strong network e¤ects, already after the second player (of 20) the majority
decision can be predicted with great reliability.
Finally, one treatment in this experiment was designed to test reputation e¤ects in the framework of a cascade model. Reputation of a player was presented as the cumulative payo¤ the player earned in previous and unrelated rounds. Subjects could observe these payo¤s, and we found strong support for the hypothesis that the decision of the player with the highest reputation signi…cantly in‡uences the choice behavior of later subjects.
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Appendix: Instructions
Once connected to our website www.A-oder-B.de, there was …rst a general overview on the experiment (screen 1 below). Then, subjects where asked to provide some personal information (screen 2 below). Only if all information was provided, subjects were allowed to continue and learn their player number as well as the monetary incentives in the current phase of the experiment (screen 3 below). Note that the number of lottery tickets and the prizes mentioned below relate to Phase I of the experiment. Subsequently, the actual experiment began. Screen 4 below provides an example of the …rst of three stages (treat-ment BHW), and we point out how these instructions were altered in case of treatments
BHW+AS and BHW+R. Screen 5 below provides an example of a treatment with payo¤ externalities played in the second stage (treatmentEarly bird). The other treatments with payo¤ externalities were explained in a similar fashion. As each stage had the same basic structure, we do not provide an example of the third stage.
Subjects also had at all times the option of opening a pop–up window that contained a summary of the main features of the set–up. All phrases emphasized in this translation were also emphasized in the original web page.
Screen 1: Introduction
A game-theoretic experiment Are you a good decision-maker? We challenge you!
Professor J. Oechssler together with the “Laboratorium for Experimental Research in Eco-nomics” at the University of Bonn aims to test various scienti…c theories through the online–experiment “A-or-B”. Financial support is provided by the consultancy McKinsey & Company.
Attractive prizes By participating in the experiment you support the scienti…c work
of the University of Bonn. At the same time you participate in a lottery for a total of 5,000 Euros which are distributed among 5 of the participants. The more thorough your decisions are, the greater your chances of winning. Of course you will also need some luck. The game takes approximately 15 minutes.
The experiment The experiment consists of three rounds. In every round you’ll be
assigned to a group and you - as well as every other member of your group - will have to take an investment decision. Without background knowledge the decision would be pure speculation. However, all players in a group will receive tips by investment bankers. Each group member gets a tip from a di¤erent investment banker. The investment bankers are experienced but can’t make perfect predictions. The reliability of the tip is the same for every investment banker. As additional information, each player can observe the decisions of his predecessors in his group.
For each correct decision you will earn a predetermined amount of Lotto–Euros. After the third round, the Lotto–Euros you earned will be converted into lottery tickets on a one-to-one basis. Hence, the better your investment decisions, the higher your chances of winning. The experiment ends on June 7, 2002. The winners of the lottery will be noti…ed after June 16, 2002 via ordinary mail. Now, let’s begin the experiment!
Screen 2: Request of personal information
Welcome to the online-experiment ”A-or-B”. Please note that you can only play once. Before the game starts, we would like to ask you for some personal information. Of course, the results of the game will be kept separately from your personal information and will be analyzed anonymously. The mail address is only needed to notify the winners. Informa-tion on your …eld of studies, age, sex, etc. are only used for scienti…c purposes. Detailed information regarding data protection may be found here [Link].
[Data entry …elds for last name, …rst name, address, email, student status, …eld of studies, year of studies, Ph.D. status, age, and sex]
Screen 3: Player number and incentives
Thank you for providing the requested information. Your player number is: [player num-ber]. Your player number, the number of lottery tickets you won, and additional informa-tion regarding the experiment will be automatically send to your email address after you have completed the experiment.
In this phase of the experiment, a total of 40,000 lottery tickets will be distributed, and 5 participants can win 1000 Euros each. Every lottery ticket has the same chance of winning.
Screen 4: Stage 1
You have to make an important investment decision: there are two risky assets (A and B). Only one asset will be successful and pay out 10 Lotto-Euros (LE). The other asset will yield no pro…t at all. The successful asset was determined randomly before the …rst player of this group played. Hence, the same asset is successful for all players in your group.
Without additional information you can rely on the fact that in55% of cases asset A is successful while in45% of casesasset B is successful.
Each participant in your group faces the same problem as you do: he has to choose between the assets and receives a tip from his respective investment banker. The reliability of the
tips is the same for all investment bankers, and the tips of the investment bankers are in-dependent of each other. The tip of each investment banker is correct in60% of the cases, i.e., in 100 cases where asset A (respectively B) is successful, in 60 cases the investment banker gives the correct tip A (respectively B) while in 40 cases the tip is not correct. The tip of your investment banker is: [B]
While each participant only knows the tip of his own investment banker, you - as every player in your group - can observe the decisions of the respective predecessors. Which players are assigned to which group is random and will di¤er from round to round. You are the [4th] investor in this group. One after another, your predecessors have made the following decisions:
Investor no. 1 2 3
Decision B A B What do you choose? [A] or [B].
Was the decision di¢ cult? Independent of your decision, what do you think is the proba-bility of A being the successful asset? [ ] %.
After the third round you’ll …nd out whether your decision was correct. Let’s move on to the next round.
[In case of treatment BHW+AS, in addition to the decisions also the tips of the pre-decessors were displayed, and the third paragraph of Screen 4 was replaced by: ”You - as every player in your group - can observe the decisions of the respective predecessors and the tips that they have received from their respective investment bankers. Which players are assigned to which group is random and will di¤er from round to round. You are the [4th] investor in this group. One after another, your predecessors have made the following decisions and have received the following tips:”].
[In case of treatment BHW+R, in addition to the decisions also the cumulative payo¤s of the predecessors earned in the respective other two stages were displayed, and the third paragraph of Screen 4 was replaced by: ”While each participant only knows the tip of his own investment banker, you - as every player in your group - can observe the decisions of the respective predecessors. In addition, each participant can observe how many Lotto– Euros their respective predecessors have earned in their respective other two stages. Which players are assigned to which group is random and will di¤er from round to round. You are the [4th] investor in this group. One after another, your predecessors have made the following decisions and have earned the following amount of Lotto–Euros on their respec-tive two other stages:”].
Another investment decision has to be made. The basic structure remains the same as in round 1. (In case you want to review the central features of round 1 please click [here].) Again, there are two risky assets (A and B). Only one asset will be successful and pay out 10 Lotto-Euros (LE). In 55% of cases it is asset A that is successful. As in the …rst round the successful asset was determined randomly before the …rst player of this group played. Hence, it is not necessarily the same asset as in the previous round that is successful.
As in round 1, every participant receives a tip from his investment banker that is cor-rect in 60% of all cases. This time, your investment banker recommends: [A]
In contrast to round 1, each participant has to pay 0.4 LE for each of his predecessors
in his group who has selectedthe same asset as himself - independent of whether his de-cision to choose A respectively B turns out to be successful, or not.
Consider the following example: suppose you were the …fth participant in a group and your predecessors had made the choices BABB. If you also would choose B, you would have to make a payment of 3 0:4LE because three of your predecessors have chosen B. If you would choose A, you would have to pay1 0:4LE.
In order to be able to make these payments you receive an endowment of 5 Lotto-Euros. Once the above payments have been deducted, you can keep the remainder.
While each participant only knows the tip of his own investment banker, you - as every player in your group - can observe the decisions of the respective predecessors. You are the [4th] investor in this group. One after another, your predecessors have made the following decisions:
Investor no. 1 2 3
Decision B A B What do you choose? [A] or [B].
Was the decision di¢ cult? Independent of your decision, what do you think is the proba-bility of A being the successful asset? [ ] %.
After the third round you’ll …nd out whether your decision was correct. Let’s move on to the next round.
